HOME

TheInfoList



OR:

The Smith chart, invented by Phillip H. Smith (1905–1987) and independently by Mizuhashi Tosaku, is a graphical calculator or nomogram designed for electrical and electronics engineers specializing in
radio frequency Radio frequency (RF) is the oscillation rate of an alternating electric current or voltage or of a magnetic, electric or electromagnetic field or mechanical system in the frequency range from around to around . This is roughly between the up ...
(RF) engineering to assist in solving problems with
transmission line In electrical engineering, a transmission line is a specialized cable or other structure designed to conduct electromagnetic waves in a contained manner. The term applies when the conductors are long enough that the wave nature of the transmi ...
s and matching circuits. The Smith chart can be used to simultaneously display multiple parameters including
impedances In electrical engineering, impedance is the opposition to alternating current presented by the combined effect of resistance and reactance in a circuit. Quantitatively, the impedance of a two-terminal circuit element is the ratio of the compl ...
,
admittance In electrical engineering, admittance is a measure of how easily a circuit or device will allow a current to flow. It is defined as the reciprocal of impedance, analogous to how conductance & resistance are defined. The SI unit of admittanc ...
s,
reflection coefficient In physics and electrical engineering the reflection coefficient is a parameter that describes how much of a wave is reflected by an impedance discontinuity in the transmission medium. It is equal to the ratio of the amplitude of the reflected ...
s, S_\,
scattering parameters Scattering parameters or S-parameters (the elements of a scattering matrix or S-matrix) describe the electrical behavior of linear electrical networks when undergoing various steady state stimuli by electrical signals. The parameters are useful f ...
, noise figure circles, constant gain contours and regions for unconditional stability, including mechanical
vibration Vibration is a mechanical phenomenon whereby oscillations occur about an equilibrium point. The word comes from Latin ''vibrationem'' ("shaking, brandishing"). The oscillations may be periodic, such as the motion of a pendulum—or random, su ...
s analysis. The Smith chart is most frequently used at or within the unity radius region. However, the remainder is still mathematically relevant, being used, for example, in
oscillator Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum ...
design and stability analysis. While the use of paper Smith charts for solving the complex mathematics involved in matching problems has been largely replaced by software based methods, the Smith chart is still a very useful method of showing how RF parameters behave at one or more frequencies, an alternative to using tabular information. Thus most RF circuit analysis software includes a Smith chart option for the display of results and all but the simplest impedance measuring instruments can plot measured results on a Smith chart display.


Overview

The Smith chart is a mathematical transformation of the two-dimensional Cartesian complex plane. Complex numbers with positive real parts map inside the circle. Those with negative real parts map outside the circle. If we are dealing only with impedances with non-negative resistive components, our interest is focused on the area inside the circle. The transformation, for an impedance Smith chart, is: \Gamma = \frac = \frac, where z = \frac, i.e., the complex impedance, Z, normalized by the reference impedance, Z_0. The impedance Smith chart is then an Argand plot of impedances thus transformed. Impedances with non-negative resistive components will appear inside a circle with unit radius; the origin will correspond to the reference impedance, Z_0. The Smith chart is plotted on the complex
reflection coefficient In physics and electrical engineering the reflection coefficient is a parameter that describes how much of a wave is reflected by an impedance discontinuity in the transmission medium. It is equal to the ratio of the amplitude of the reflected ...
plane in two dimensions and may be scaled in normalised impedance (the most common), normalised
admittance In electrical engineering, admittance is a measure of how easily a circuit or device will allow a current to flow. It is defined as the reciprocal of impedance, analogous to how conductance & resistance are defined. The SI unit of admittanc ...
or both, using different colours to distinguish between them. These are often known as the Z, Y and YZ Smith charts respectively. Normalised scaling allows the Smith chart to be used for problems involving any characteristic or system impedance which is represented by the center point of the chart. The most commonly used normalization impedance is 50 
ohm Ohm (symbol Ω) is a unit of electrical resistance named after Georg Ohm. Ohm or OHM may also refer to: People * Georg Ohm (1789–1854), German physicist and namesake of the term ''ohm'' * Germán Ohm (born 1936), Mexican boxer * Jörg Ohm (bor ...
s. Once an answer is obtained through the graphical constructions described below, it is straightforward to convert between normalised impedance (or normalised admittance) and the corresponding unnormalized value by multiplying by the characteristic impedance (admittance). Reflection coefficients can be read directly from the chart as they are unitless parameters. The Smith chart has a scale around its
circumference In geometry, the circumference (from Latin ''circumferens'', meaning "carrying around") is the perimeter of a circle or ellipse. That is, the circumference would be the arc length of the circle, as if it were opened up and straightened out ...
or periphery which is graduated in
wavelengths In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tr ...
and degrees. The wavelengths scale is used in distributed component problems and represents the distance measured along the transmission line connected between the generator or source and the load to the point under consideration. The degrees scale represents the angle of the voltage reflection coefficient at that point. The Smith chart may also be used for lumped-element matching and analysis problems. Use of the Smith chart and the interpretation of the results obtained using it requires a good understanding of AC circuit theory and transmission-line theory, both of which are prerequisites for RF engineers. As impedances and admittances change with frequency, problems using the Smith chart can only be solved manually using one
frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
at a time, the result being represented by a point. This is often adequate for narrow band applications (typically up to about 5% to 10% bandwidth) but for wider bandwidths it is usually necessary to apply Smith chart techniques at more than one frequency across the operating frequency band. Provided the frequencies are sufficiently close, the resulting Smith chart points may be joined by straight lines to create a locus. A locus of points on a Smith chart covering a range of frequencies can be used to visually represent: *how capacitive or how inductive a load is across the frequency range *how difficult matching is likely to be at various frequencies *how well matched a particular component is. The accuracy of the Smith chart is reduced for problems involving a large locus of impedances or admittances, although the scaling can be magnified for individual areas to accommodate these.


Mathematical basis


Actual and normalised impedance and admittance

A transmission line with a characteristic impedance of Z_0\, may be universally considered to have a characteristic admittance of Y_0\, where :Y_0 = \frac\, Any impedance, Z_\text\, expressed in ohms, may be normalised by dividing it by the characteristic impedance, so the normalised impedance using the lower case ''z''T is given by :z_\text = \frac\, Similarly, for normalised admittance :y_\text = \frac\, The
SI unit The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric system and the world's most widely used system of measurement. ...
of impedance is the
ohm Ohm (symbol Ω) is a unit of electrical resistance named after Georg Ohm. Ohm or OHM may also refer to: People * Georg Ohm (1789–1854), German physicist and namesake of the term ''ohm'' * Germán Ohm (born 1936), Mexican boxer * Jörg Ohm (bor ...
with the symbol of the upper case
Greek letter The Greek alphabet has been used to write the Greek language since the late 9th or early 8th century BCE. It is derived from the earlier Phoenician alphabet, and was the earliest known alphabetic script to have distinct letters for vowels as ...
omega Omega (; capital: Ω, lowercase: ω; Ancient Greek ὦ, later ὦ μέγα, Modern Greek ωμέγα) is the twenty-fourth and final letter in the Greek alphabet. In the Greek numeric system/ isopsephy ( gematria), it has a value of 800. The ...
(Ω) and the
SI unit The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric system and the world's most widely used system of measurement. ...
for
admittance In electrical engineering, admittance is a measure of how easily a circuit or device will allow a current to flow. It is defined as the reciprocal of impedance, analogous to how conductance & resistance are defined. The SI unit of admittanc ...
is the
siemens Siemens AG ( ) is a German multinational conglomerate corporation and the largest industrial manufacturing company in Europe headquartered in Munich with branch offices abroad. The principal divisions of the corporation are ''Industry'', ''E ...
with the symbol of an upper case letter S. Normalised impedance and normalised admittance are
dimensionless A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...
. Actual impedances and admittances must be normalised before using them on a Smith chart. Once the result is obtained it may be de-normalised to obtain the actual result.


The normalised impedance Smith chart

Using transmission-line theory, if a transmission line is terminated in an impedance (Z_\text\,) which differs from its characteristic impedance (Z_0\,), a
standing wave In physics, a standing wave, also known as a stationary wave, is a wave that oscillates in time but whose peak amplitude profile does not move in space. The peak amplitude of the wave oscillations at any point in space is constant with respect ...
will be formed on the line comprising the
resultant In mathematics, the resultant of two polynomials is a polynomial expression of their coefficients, which is equal to zero if and only if the polynomials have a common root (possibly in a field extension), or, equivalently, a common factor (ov ...
of both the incident or forward (V_\text\,) and the reflected or reversed (V_\text\,) waves. Using complex
exponential Exponential may refer to any of several mathematical topics related to exponentiation, including: *Exponential function, also: **Matrix exponential, the matrix analogue to the above *Exponential decay, decrease at a rate proportional to value *Expo ...
notation: :V_\text = A \exp(j \omega t)\exp(+\gamma \ell)~\, and :V_\text = B \exp(j \omega t)\exp(-\gamma \ell)\, where :\exp(j \omega t)\, is the temporal part of the wave :\exp(\pm\gamma \ell)\, is the spatial part of the wave and :\omega = 2 \pi f\, where :\omega\, is the
angular frequency In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit ti ...
in
radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that ...
s per
second The second (symbol: s) is the unit of time in the International System of Units (SI), historically defined as of a day – this factor derived from the division of the day first into 24 hours, then to 60 minutes and finally to 60 seconds ea ...
(rad/s) :f\, is the
frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
in
hertz The hertz (symbol: Hz) is the unit of frequency in the International System of Units (SI), equivalent to one event (or cycle) per second. The hertz is an SI derived unit whose expression in terms of SI base units is s−1, meaning that o ...
(Hz) :t\, is the time in seconds (s) :A\, and B\, are constants :\ell\, is the distance measured along the transmission line from the load toward the generator in metres (m) Also :\gamma = \alpha + j \beta\, is the
propagation constant The propagation constant of a sinusoidal electromagnetic wave is a measure of the change undergone by the amplitude and phase of the wave as it propagates in a given direction. The quantity being measured can be the voltage, the current in a c ...
which has
SI units The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric system and the world's most widely used system of measurement. ...
radians The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that ...
/
meter The metre (British spelling) or meter (American spelling; see spelling differences) (from the French unit , from the Greek noun , "measure"), symbol m, is the primary unit of length in the International System of Units (SI), though its pref ...
where :\alpha\, is the attenuation constant in
neper The neper (symbol: Np) is a logarithmic unit for ratios of measurements of physical field and power quantities, such as gain and loss of electronic signals. The unit's name is derived from the name of John Napier, the inventor of logarithms. A ...
s per metre (Np/m) :\beta\, is the phase constant in
radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that ...
s per metre (rad/m) The Smith chart is used with one frequency (\omega) at a time, and only for one moment (t) at a time, so the temporal part of the phase (\exp(j \omega t)\,) is fixed. All terms are actually multiplied by this to obtain the
instantaneous phase Instantaneous phase and frequency are important concepts in signal processing that occur in the context of the representation and analysis of time-varying functions. The instantaneous phase (also known as local phase or simply phase) of a ''comple ...
, but it is conventional and understood to omit it. Therefore, :V_\text = A \exp(+\gamma \ell)\, and :V_\text = B \exp(-\gamma \ell)\, where A\, and B\, are respectively the forward and reverse voltage amplitudes at the load.


The variation of complex reflection coefficient with position along the line

The complex voltage reflection coefficient \Gamma\, is defined as the ratio of the reflected wave to the incident (or forward) wave. Therefore, :\Gamma = \frac = \frac = C \exp(-2 \gamma \ell)\, where is also a constant. For a uniform transmission line (in which \gamma\, is constant), the complex reflection coefficient of a standing wave varies according to the position on the line. If the line is
lossy In information technology, lossy compression or irreversible compression is the class of data compression methods that uses inexact approximations and partial data discarding to represent the content. These techniques are used to reduce data si ...
(\alpha\, is non-zero) this is represented on the Smith chart by a
spiral In mathematics, a spiral is a curve which emanates from a point, moving farther away as it revolves around the point. Helices Two major definitions of "spiral" in the American Heritage Dictionary are:Möbius transformation In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form f(z) = \frac of one complex variable ''z''; here the coefficients ''a'', ''b'', ''c'', ''d'' are complex numbers satisfying ''ad' ...
. Both \,\Gamma\, and \,z_\mathsf\, are expressed in
complex numbers In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
without any units. They both change with frequency so for any particular measurement, the frequency at which it was performed must be stated together with the characteristic impedance. \,\Gamma\, may be expressed in magnitude and
angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles ...
on a polar diagram. Any actual reflection coefficient must have a magnitude of less than or equal to
unity Unity may refer to: Buildings * Unity Building, Oregon, Illinois, US; a historic building * Unity Building (Chicago), Illinois, US; a skyscraper * Unity Buildings, Liverpool, UK; two buildings in England * Unity Chapel, Wyoming, Wisconsin, US; a ...
so, at the test frequency, this may be expressed by a point inside a circle of unity radius. The Smith chart is actually constructed on such a polar diagram. The Smith chart scaling is designed in such a way that reflection coefficient can be converted to normalised impedance or vice versa. Using the Smith chart, the normalised impedance may be obtained with appreciable accuracy by plotting the point representing the reflection coefficient ''treating the Smith chart as a polar diagram'' and then reading its value directly using the characteristic Smith chart scaling. This technique is a graphical alternative to substituting the values in the equations. By substituting the expression for how reflection coefficient changes along an unmatched loss-free transmission line : \Gamma = \frac = \frac \, for the loss free case, into the equation for normalised impedance in terms of reflection coefficient : z_\mathsf = \frac \,. and using
Euler's formula Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that ...
: \exp(j\theta) = \cos \theta + j\, \sin \theta \, yields the impedance-version transmission-line equation for the loss free case: :Z_\mathsf = Z_0 \frac \, where \,Z_\mathsf\, is the impedance 'seen' at the input of a loss free transmission line of length \,\ell\, , terminated with an impedance \,Z_\mathsf\, Versions of the transmission-line equation may be similarly derived for the admittance loss free case and for the impedance and admittance lossy cases. The Smith chart graphical equivalent of using the transmission-line equation is to normalise \, Z_\mathsf \, , to plot the resulting point on a Smith chart and to draw a circle through that point centred at the Smith chart centre. The path along the arc of the circle represents how the impedance changes whilst moving along the transmission line. In this case the circumferential (wavelength) scaling must be used, remembering that this is the wavelength within the transmission line and may differ from the free space wavelength.


Regions of the Smith chart

If a polar diagram is mapped on to a
cartesian coordinate system A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
it is conventional to measure angles relative to the positive -axis using a
counterclockwise Two-dimensional rotation can occur in two possible directions. Clockwise motion (abbreviated CW) proceeds in the same direction as a clock's hands: from the top to the right, then down and then to the left, and back up to the top. The opposite ...
direction for positive angles. The magnitude of a complex number is the length of a straight line drawn from the
origin Origin(s) or The Origin may refer to: Arts, entertainment, and media Comics and manga * Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002 * The Origin (Buffy comic), ''The Origin'' (Bu ...
to the point representing it. The Smith chart uses the same convention, noting that, in the normalised impedance plane, the positive -axis extends from the center of the Smith chart at \, z_\mathsf = 1 \pm j 0 \, to the point \, z_\mathsf = \infty \pm j \infty \,. The region above the x-axis represents inductive impedances (positive imaginary parts) and the region below the -axis represents capacitive impedances (negative imaginary parts). If the termination is perfectly matched, the reflection coefficient will be zero, represented effectively by a circle of zero radius or in fact a point at the centre of the Smith chart. If the termination was a perfect open circuit or
short circuit A short circuit (sometimes abbreviated to short or s/c) is an electrical circuit that allows a current to travel along an unintended path with no or very low electrical impedance. This results in an excessive current flowing through the circu ...
the magnitude of the reflection coefficient would be unity, all power would be reflected and the point would lie at some point on the unity circumference circle.


Circles of constant normalised resistance and constant normalised reactance

The normalised impedance Smith chart is composed of two families of circles: circles of constant normalised resistance and circles of constant normalised reactance. In the complex reflection coefficient plane the Smith chart occupies a circle of unity radius centred at the origin. In cartesian coordinates therefore the circle would pass through the points (+1,0) and (−1,0) on the -axis and the points (0,+1) and (0,−1) on the -axis. Since both \,\Gamma\, and \,z_\mathsf\, are complex numbers, in general they may be written as: :z_\mathsf = a + j b \, :~ \Gamma ~= c + j d \, with ''a'', ''b'', ''c'' and ''d'' real numbers. Substituting these into the equation relating normalised impedance and complex reflection coefficient: :\Gamma = \frac = \frac \, gives the following result: :\Gamma = c + j d = \left frac\right+ j \left frac\right = \left 1 + \frac\right+ j \left frac\right\,. This is the equation which describes how the complex reflection coefficient changes with the normalised impedance and may be used to construct both families of circles.


The Smith chart

The Smith chart is constructed in a similar way to the Smith chart case but by expressing values of voltage reflection coefficient in terms of normalised admittance instead of normalised impedance. The normalised admittance is the reciprocal of the normalised impedance , so : y_\mathsf = \frac \, Therefore: : y_\mathsf = \frac \, and : \Gamma = \frac \, The Smith chart appears like the normalised impedance, type but with the graphic nested circles rotated through 180°, but the numeric scale remaining in its same position (not rotated) as the chart. Similarly taking : y_\mathsf = \tilde + j\,\tilde \, for
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
\,\tilde\, and \,\tilde\, gives an analogous result, although with more and different minus signs: :\Gamma = c + j d = \left frac\right+ j \left frac\right = \left \frac- 1 \right+ j \left frac\right\,. The region above the -axis represents capacitive admittances and the region below the -axis represents inductive admittances. Capacitive admittances have positive imaginary parts and inductive admittances have negative imaginary parts. Again, if the termination is perfectly matched the reflection coefficient will be zero, represented by a 'circle' of zero radius or in fact a point at the centre of the Smith chart. If the termination was a perfect open or short circuit the magnitude of the voltage reflection coefficient would be unity, all power would be reflected and the point would lie at some point on the unity circumference circle of the Smith chart.


Practical examples

A point with a reflection coefficient magnitude 0.63 and angle 60° represented in polar form as 0.63\angle60^\circ\,, is shown as point P1 on the Smith chart. To plot this, one may use the circumferential (reflection coefficient) angle scale to find the \angle60^\circ\, graduation and a ruler to draw a line passing through this and the centre of the Smith chart. The length of the line would then be scaled to P1 assuming the Smith chart radius to be unity. For example, if the actual radius measured from the paper was 100 mm, the length OP1 would be 63 mm. The following table gives some similar examples of points which are plotted on the ''Z'' Smith chart. For each, the reflection coefficient is given in polar form together with the corresponding normalised impedance in rectangular form. The conversion may be read directly from the Smith chart or by substitution into the equation.


Working with both the ''Z'' Smith chart and the ''Y'' Smith charts

In RF circuit and matching problems sometimes it is more convenient to work with admittances (representing conductances and
susceptance In electrical engineering, susceptance (''B'') is the imaginary part of admittance, where the real part is conductance. The reciprocal of admittance is impedance, where the imaginary part is reactance and the real part is resistance. In SI uni ...
s) and sometimes it is more convenient to work with impedances (representing resistances and reactances). Solving a typical matching problem will often require several changes between both types of Smith chart, using normalised impedance for series elements and normalised admittances for
parallel Parallel is a geometric term of location which may refer to: Computing * Parallel algorithm * Parallel computing * Parallel metaheuristic * Parallel (software), a UNIX utility for running programs in parallel * Parallel Sysplex, a cluster o ...
elements. For these a dual (normalised) impedance and admittance Smith chart may be used. Alternatively, one type may be used and the scaling converted to the other when required. In order to change from normalised impedance to normalised admittance or vice versa, the point representing the value of reflection coefficient under consideration is moved through exactly 180 degrees at the same radius. For example, the point P1 in the example representing a reflection coefficient of 0.63\angle60^\circ\, has a normalised impedance of z_P = 0.80 + j1.40\,. To graphically change this to the equivalent normalised admittance point, say Q1, a line is drawn with a ruler from P1 through the Smith chart centre to Q1, an equal radius in the opposite direction. This is equivalent to moving the point through a circular path of exactly 180 degrees. Reading the value from the Smith chart for Q1, remembering that the scaling is now in normalised admittance, gives y_P = 0.30 - j0.54\,. Performing the calculation :y_\text = \frac\, manually will confirm this. Once a transformation from impedance to admittance has been performed, the scaling changes to normalised admittance until a later transformation back to normalised impedance is performed. The table below shows examples of normalised impedances and their equivalent normalised admittances obtained by rotation of the point through 180°. Again, these may be obtained either by calculation or using a Smith chart as shown, converting between the normalised impedance and normalised admittances planes.


Choice of Smith chart type and component type

The choice of whether to use the ''Z'' Smith chart or the ''Y'' Smith chart for any particular calculation depends on which is more convenient. Impedances in series and admittances in parallel add while impedances in parallel and admittances in series are related by a reciprocal equation. If Z_\text is the equivalent impedance of series impedances and Z_\text is the equivalent impedance of parallel impedances, then :Z_\text = Z_1 + Z_2 + Z_3 + ... \, :\frac = \frac + \frac + \frac + ... \, For admittances the reverse is true, that is :Y_\text = Y_1 + Y_2 + Y_3 + ... \, :\frac = \frac + \frac + \frac + ... \, Dealing with the reciprocals, especially in complex numbers, is more time consuming and error-prone than using linear addition. In general therefore, most RF engineers work in the plane where the circuit topography supports linear addition. The following table gives the complex expressions for impedance (real and normalised) and admittance (real and normalised) for each of the three basic passive circuit elements: resistance, inductance and capacitance. Using just the characteristic impedance (or characteristic admittance) and test frequency an equivalent circuit can be found and vice versa.


Using the Smith chart to solve conjugate matching problems with distributed components

Distributed matching becomes feasible and is sometimes required when the physical size of the matching components is more than about 5% of a wavelength at the operating frequency. Here the electrical behaviour of many lumped components becomes rather unpredictable. This occurs in microwave circuits and when high power requires large components in shortwave, FM and TV Broadcasting, For distributed components the effects on reflection coefficient and impedance of moving along the transmission line must be allowed for using the outer circumferential scale of the Smith chart which is calibrated in wavelengths. The following example shows how a transmission line, terminated with an arbitrary load, may be matched at one frequency either with a series or parallel reactive component in each case connected at precise positions. Supposing a loss-free air-spaced transmission line of characteristic impedance Z_0 = 50 \ \Omega, operating at a frequency of 800 MHz, is terminated with a circuit comprising a 17.5 \Omega resistor in series with a 6.5 nanohenry (6.5 nH) inductor. How may the line be matched? From the table above, the reactance of the inductor forming part of the termination at 800 MHz is :Z_L = j \omega L = j 2 \pi f L = j32.7 \ \Omega\, so the impedance of the combination (Z_T) is given by :Z_T = 17.5 + j32.7 \ \Omega\, and the normalised impedance (z_T) is :z_T = \frac = 0.35 + j0.65\, This is plotted on the Z Smith chart at point P20. The line OP20 is extended through to the wavelength scale where it intersects at the point L_1 = 0.098 \lambda\,. As the transmission line is loss free, a circle centred at the centre of the Smith chart is drawn through the point P20 to represent the path of the constant magnitude reflection coefficient due to the termination. At point P21 the circle intersects with the unity circle of constant normalised resistance at :z_ = 1.00 + j1.52\,. The extension of the line OP21 intersects the wavelength scale at L_2 = 0.177 \lambda\,, therefore the distance from the termination to this point on the line is given by :L_2 - L_1 = 0.177\lambda - 0.098\lambda = 0.079\lambda\, Since the transmission line is air-spaced, the wavelength at 800 MHz in the line is the same as that in free space and is given by :\lambda = \frac\, where c\, is the velocity of electromagnetic radiation in free space and f\, is the frequency in hertz. The result gives \lambda = 375 \ \mathrm\,, making the position of the matching component 29.6 mm from the load. The conjugate match for the impedance at P21 (z_\,) is :z_ = - j (1.52),\! As the Smith chart is still in the normalised impedance plane, from the table above a series capacitor C_m\, is required where :z_ = - j 1.52 = \frac = \frac\, Rearranging, we obtain :C_m=\frac = \frac. Substitution of known values gives :C_m = 2.6 \ \mathrm\, To match the termination at 800 MHz, a series capacitor of 2.6 pF must be placed in series with the transmission line at a distance of 29.6 mm from the termination. An alternative shunt match could be calculated after performing a Smith chart transformation from normalised impedance to normalised admittance. Point Q20 is the equivalent of P20 but expressed as a normalised admittance. Reading from the Smith chart scaling, remembering that this is now a normalised admittance gives :y_ = 0.65 - j1.20\, (In fact this value is not actually used). However, the extension of the line OQ20 through to the wavelength scale gives L_3 = 0.152 \lambda\,. The earliest point at which a shunt conjugate match could be introduced, moving towards the generator, would be at Q21, the same position as the previous P21, but this time representing a normalised admittance given by :y_ = 1.00 + j1.52\,. The distance along the transmission line is in this case :L_2 + L_3 = 0.177\lambda + 0.152\lambda = 0.329\lambda\, which converts to 123 mm. The conjugate matching component is required to have a normalised admittance (y_) of :y_ = - j1.52\,. From the table it can be seen that a negative admittance would require an inductor, connected in parallel with the transmission line. If its value is L_m\,, then :-j1.52 = \frac= \frac\, This gives the result :L_m = 6.5 \ \mathrm\, A suitable inductive shunt matching would therefore be a 6.5 nH inductor in parallel with the line positioned at 123 mm from the load.


3D Smith chart

A generalization of the Smith chart to a three dimensional sphere, based on the extended complex plane (
Riemann sphere In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers ...
) and
inversive geometry Inversive activities are processes which self internalise the action concerned. For example, a person who has an Inversive personality internalises his emotion Emotions are mental states brought on by neurophysiological changes, variou ...
, was proposed by Muller, ''et al'' in 2011. The chart unifies the passive and active circuit design on little and big circles on the surface of a unit sphere, using a stereographic
conformal map In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths. More formally, let U and V be open subsets of \mathbb^n. A function f:U\to V is called conformal (or angle-preserving) at a point u_0\in ...
of the reflection coefficient's generalized plane. Considering the point at infinity, the space of the new chart includes all possible loads: The north pole is the perfectly matched point, while the south pole is the completely mismatched point. The 3D Smith chart has been further extended outside of the spherical surface, for plotting various scalar parameters, such as group delay, quality factors, or frequency orientation. The visual frequency orientation (clockwise vs. counter-clockwise) enables one to differentiate between a negative / capacitance and positive / inductive whose reflection coefficients are the same when plotted on a 2D Smith chart, but whose orientations diverge as frequency increases.


References


Further reading

* For an early representation of this graphical depiction before they were called 'Smith Charts', see , In particular, Fig. 13 on p. 810.


External links

* * * * Non-commercial, interactive Smith Chart that looks best in Excel 2007+. * Non-commercial, available for Windows, Mac, and Linux. Many Smith chart tutorial videos. No circuit size restrictions. Not limited to ladder circuits. * Commercial and free Smith chart for Windows * Free web based Smith Chart Educational tool available on
GitHub GitHub, Inc. () is an Internet hosting service for software development and version control using Git. It provides the distributed version control of Git plus access control, bug tracking, software feature requests, task management, cont ...
. * 2D and 3D Smith chart generalized tool for active and passive circuits (free for academia/education). {{Authority control Electrical engineering Charts